You see, folks, there are these things called Johnson Solids. To simplify, they are polyhedra made of regular polygons in irregular configurations.
Some time ago, I attempted to produce the fourth-dimensional equivalents of a few of them by using methods of constructing them that didn't rely on their given dimension. For example, instead of constructing a triangular cupola by connecting a polygon to a polygon with twice as many sides, you take a polygon and extend it's sides outward to form new faces. For the tetrahedral cupola, the faces are extended outward, becoming triangular prisms. If I were to attempt to take a tetrahedron and attach it's faces to a polyhedron with twice as many sides it would be irregular. If I merely swapped all the faces for their 3D counterparts, they would not fit together.
While this was a fun exercise, it was futile, and I soon discovered that an algorithmic search for 4D Johnson Solids was already in place, with many thousands of shapes already discovered. Whether or not the shapes I had noted fit the qualifications of a true 4D Johnson I still do not know, nor even if they were all possible. However, I had already made these few renders, before giving up on the idea.
However, as of July 2009 I've decided the effort may not be entirely futile, for two reasons.
As of the 29th of July 2010, I have derived another analogue, this time for one of the elementary Johnsons. I will detail the solid and my method for derivation at next opportunity. My method may also be helpful for deriving further elementaries.
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